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Complex Equation Conclusion (Posted on 2015-12-12) Difficulty: 3 of 5
A and B are complex numbers such that:

A2 + B2 = 7
A3 + B3 = 10

What is the maximum possible real value of A + B?

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution | Comment 1 of 3

Changing variables to u and v where u = A + B and v = A – B,

i.e. substituting A = (u + v)/2 and  B = (u – v)/2 :

    (u + v)2/4 + (u – v)2/4 = 7   which gives        u2 + v2 = 14  (1)

& (u + v)3/8 + (u – v)3/8 = 10  which gives  u(u2 + 3v2) = 40  (2)

From (1), v2 = 14 – u2, which can now be substituted into (2):

            u(u2 + 42 – 3u2) = 40

                 u3 -21u + 20 = 0

     (u – 1)(u – 4)(u + 5) = 0  giving         a + b = u = 1, 4, -5

Was I alone in wondering whether ‘maximum real value of a + b’
was meant to be ‘maximum real part of a + b’? All is well, it turns
out that a + b can only have real values, and the maximum is 4.



  Posted by Harry on 2015-12-12 15:48:36
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